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An Atlas of Functions: With Equator, The Atlas Function Calculator

Keith Oldham, Jan Myland, and Jerome Spanier
Publication Date: 
Number of Pages: 
Hardcover with CDROM
[Reviewed by
John Cook
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An Atlas of Functions by Keith Oldham, Jan Myland, and Jerome Spanier is a heavy book, metaphorically and literally. It contains a wealth of information and weighs in at 5.1 lbs (2.3 kg). The book is primarily concerned with special functions. The title omits the word “special” for a good reason: some of the functions in the book are not “special” in the traditional sense.

There's no rigorous definition special functions, but the following definition is in line with the general consensus: functions that are commonly used in applications, have many nice properties, and are not typically available on a calculator. Obviously this is a sloppy definition, and yet it works fairly well in practice. Most people would agree, for example, that the gamma function is included in the list of special functions and that trigonometric functions are not.

The Atlas includes chapters for several functions that are not “special.” About 20% of the book is devoted to functions such as quadratic and cubic polynomials, absolute value, square roots, etc. These functions are commonly used in applications and have many nice properties, but they are classified as “elementary” rather than “special” because they can be computed on a typical calculator. However, the chapters on “elementary” functions are not as elementary as one might expect. The book examines even simple functions in unusual depth.

Every chapter is divided into the same fourteen sections:

  1. Notation
  2. Behavior
  3. Definitions
  4. Special cases
  5. Intra-relationships
  6. Expansions
  7. Particular values
  8. Numerical values
  9. Limits and approximations
  10. Operations of the calculus
  11. Complex argument
  12. Generalizations
  13. Cognate functions
  14. Related topics

Most of the section titles are self-explanatory. However, three sections deserve further explanation. The sections on “operations of the calculus” include not only ordinary differentiation and integration but also fractional order derivatives and integrals. (Fractional calculus is used frequently throughout the book to establish connections between different families of functions.) In each chapter, “cognate” functions are functions related to the chapter topic but not listed under special cases or generalizations. Finally, the “related topics” given in the final sections of each chapter are quite broad. For example, some of the topics included in the final sections of chapters include cubic splines, linear regression, and Laplace transforms. Some chapters actually have 15 sections because there are two separate sections on related topics.

The Atlas devotes a considerable amount of space to exploring the connections between families of functions. Having the same internal structure for every chapter helps bring out these connections. Hypergeometric functions are mentioned frequently throughout the book because the theory of such functions clarifies relationships between groups of functions. While many books on special functions emphasize recurrence relations and differential equations as unifying principles, the Atlas places more emphasis on hypergeometric functions and operations for synthesizing one such function from another.


The most popular reference for special functions over the last 40 years has been Handbook of Mathematical Functions by Abramowitz and Stegun, affectionately known as A&S, and so it is natural to ask how the Atlas compares to this classic reference.

Technology has clearly changed since A&S was first published. Whereas A&S includes tables of function values and numerical algorithms, the Atlas comes with software for computing the functions described in the book. And while A&S is printed in black and white on matte paper; the Atlas is printed in color on glossy paper.

Atlas and A&S cover essentially the same list of special functions. (Occasionally the Atlas will use a different name than A&S for the same function. However, when a function is commonly known by multiple names, the Atlas lists the alternatives.) The Atlas goes into greater theoretical depth than A&S. While the Atlas is valuable as a reference, it also contains a large amount of expository text. A&S was written to be pulled off the shelf for quick reference. The Atlas seems to have been written for the reader who is not in such a hurry and who might enjoy looking around a bit.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.

Preface.- The Constant Function c.- The Factorial Function n!.- The Zeta Numbers and Related Functions.- The Bernoulli Numbers Bn.- The Euler Numbers En.- The Bionmial Coefficients.- The Linear Function bx + c and Its Reciprocal.- Modifying Functions.- The Heaviside and Dirac Functions.- The Integer Powers xn and (bx + c)n.- The Square-Root Function and Its Reciprocal.- The Noninteger Power xv.- The Semielliptic Function and Its Reciprocal.- The (b/a)square root of x2 +- a2 Functions and Their Reciprocals.- The Quadratic Function ax + bx + c and Its Reciprocal.- The Cubic Function x3 + bx + c.- Polynomial Functions.- The Pochhammer Polynomials (x)n.- The Bernoulli Polynomials Bn(x).- The Euler Polynomials En(x).- The Legendre Polynomials Pn(x).- The Chebyshev Polynomials Tn(x) and Un(x).- The Laguerre Polynomials Ln(x).- The Hermite Polynomials Hn(x).- The Logarithmic Function ln(x).- The Exponential Function exp(x).- Exponential of Powers.- The Hyperbolic Cosine cosh(x). and Sine sinh(x) Functions.- The Hyperbolic Secant and Cosecant Functions.- The Inverse Hyperbolic Functions.- The Cosine cox(x) and Sine sin(x) Functions.- The Secant sec(x) and Cosecant csc(x) Fucntions.- The Tangent tan(x) and Cotangent cot(x) Functions.- The Inverse Circular Functions.- Periodic Functions.- The Exponential Integrals Ei(x) and Ein(x).- Sine and Cosine Integrals.- The Fresnel Integrals C(x) and S(x).- The Error Function erf(x) and Its Complement erfc(x).- The exp(x)erfc(square root of x) and Related Functions.- Dawson's Integral daw(x).- The Gamma Function.- The Digamma Function.- The Incomplete Gamma Functions.- The Parabolic Cylinder Function Dv(x).- The Kummer Function M(a, c, x).- The Tricomi Function U(a, c, x).- The Modified Bessel Functions In(x) of Integer Order.- The Modified Bessel Functions of In(x) Arbitrary Order.- The Macdonald Function Kv(x).- The Bessel Functions Jn(x) of Integer Order.- The Bessel Functions Jv(x) of Arbitrary Order.- The Neumann Function Yv(x). The Kelvin Functions.- The Airy Functions Ai(x) and Bi(x).- The Struve Function hv(x).- The Incomplete Beta Function.- The Legendre Functions Pv(x) and Qv(x).- The Gauss Hypergeometric Function F(a,b,c,x).- The Complete Elliptic Integrals K(k) and E(k).- The Incomplete Elliptic Integrals.- The Jacobian Elliptic Functions.- The Hurwitz Function.- Appendix A: Useful Data.- Appendix B: Bibliography.- Appendix C: Equator, The Atlas Function Calculator.- Symbol Index.- Subject Index.